# Ivars Peterson's MathTrek

June 19, 2006

## Folding Perfect Thirds

Tom Hull started making origami figures when he was 8 years old. By the time he was in graduate school at the University of Rhode Island, he had found a way to combine his passion for origami with a career in mathematics. Now at Merrimack College in North Andover, Mass., he is a member of a small but growing community of origami enthusiasts intent on pursuing the mathematical intricacies of their craft.

In his efforts to collect everything that he could find linking origami and math (and in his own research efforts), Hull has discovered not only the obvious links between origami and geometry but also intriguing intersections of origami with other fields of mathematics, such as algebra, number theory, and combinatorics.

In his new book, Project Origami, Hull presents 22 hands-on activities that cover a broad swath of mathematics, ranging from folding equilateral triangles in a square to exploring Gaussian curvature. "My goal was to compile many of the origami-math aspects that I had found and present them in a way that would be easy for college or advanced high-school teachers to use in their classes," he writes.

Here's one of the simpler assignments: Fold a square into perfect thirds.

Most people have no trouble folding a sheet of paper in half, or in quarters or eighths. Folding it into thirds is a little trickier. The following set of diagrams shows how it can be done with a square sheet.

Using a square piece of paper, make two creases by folding over from side to side and along a diagonal (left). Make a third crease that extends from one corner to the midpoint of one side (middle). Where the third crease meets the diagonal crease defines a point P that is on a vertical line dividing the sheet in thirds. A new crease that passes vertically through this point defines the boundary of one-third of the sheet (right, yellow).

It's fairly easy to prove geometrically or analytically that this crease pattern truly divides the sheet into thirds.

Hull then asks how this method can be generalized to make perfect fifths or nths, where n is an odd number.

There are lots more such activities, all instructive and fun, in Hull's book.

"One of the main attractions of using origami to teach math is that it requires hands-on participation," Hull says. "There's no chance of someone hiding in the back of the room or falling asleep when everyone is trying to fold a hyperbolic paraboloid."

"The fact that origami is, by definition, hands-on makes it a natural for active learning," he continues. "One could even make the argument that while folding paper, especially when making geometric models, latent mathematical learning will always happen. There's no way a student can make a dodecahedron out of thirty PhiZZ units without an understanding of some fundamental properties of this object."

Check out Ivars Peterson's MathTrek blog at http://blog.sciencenews.org/. His Web page is at http://ivarspeterson.googlepages.com/.

References:

Hull, T. 2006. Project Origami: Activities for Exploring Mathematics. Wellesley, Mass.: A K Peters. See http://www.akpeters.com/product.asp?ProdCode=2582and http://www.akpeters.com/ProjectOrigami/.

Peterson, I. 2005.A cabinet of mathematical curiosities. MAA Online (Dec. 26).

______. 2005. Paper bags and tricky folds. MAA Online (July 25).

______. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. New York: Wiley.

______. 2001. Fancy folding. Muse 5(March):24. Available at http://www.sciencenewsforkids.org/pages/puzzlezone/muse/muse0301.asp.

______. 2001. Folding maps. MAA Online (Jan. 8).

Tom Hull has a Web site at http://www.merrimack.edu/~thull/.

For more on origami and math, go to http://www.paperfolding.com/math/.